My first suggestion is to define your exact financial parameters. Period. What is the upper limit of your budget. For example, Case Western would probably give you $30,000 for your stats. But that leaves a $30,000 gap. If your upper limit is $10-15,000, it is irrelevant that they offered $30,000.
Once you know the finacial prism you have to investigate schools through, do so wisely. Use NPC to ensure you have several schools you can be happy attending. Then start looking at the super competitive scholarships at individual schools. Apply for some of them. You can’t win them without trying, but knowing that 1-3 out of a large number of applicants will win means needing to be realistic.
One of the drawbacks our ds found last yr was that many scholarships are tied to diversity or to amazing levels of community action service. He was an active volunteer his whole life bc he is a compassionate person who always wants to help others. But, no, he didn’t change his community in a resume defining moment. Many kids have huge levels of life-altering community impact. They are the kids you will be competing against. Or if tied to awards/research, you compete against Intel winners, etc.
So you need to apply, but you also need to be realistic.
If $$ is a controlling factor, there are lower ranked schools which will offer great merit money. Many have honors programs within the university which will be filled with peers just like you describe.